Question

Find the value of $4tan^{-1} \frac{1}{5}-tan^{-1}\frac{1}{239}$

Answer 1

$4 {tan}^{ - 1} \frac{1}{5 } - {tan}^{ - 1} \frac{1}{239}$

$= 2 \times 2{tan}^{ - 1} \frac{1}{5} - {tan}^{ - 1} \frac{1}{239}$

$= 2 \times {tan}^{ - 1} \frac{ \frac{2 }{5} }{1 - ( { \frac{1}{5} )}^{2} } - {tan}^{ - 1} \frac{1}{239}$

We know, $2{tan}^{ - 1}x = {tan}^{ - 1} \frac{2x}{1 - {x}^{2} }$

$= 2{tan}^{ - 1} \frac{5}{12} - {tan}^{ - 1} \frac{1}{239}$

$= {tan}^{ - 1} \frac{2 \times \frac{5}{12} }{1 - ( { \frac{5}{12}) }^{2} } - {tan}^{ - 1} \frac{1}{239}$

$= {tan}^{ - 1} \frac{120}{119} - {tan}^{ - 1} \frac{1}{239}$

$= {tan}^{ - 1} \frac{ \frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{119} \times \frac{1}{239} }$

$= {tan}^{ - 1}1$

$= \frac{\pi}{4}$